Answer Key to accompany

Transcription

1 to accompany The Cryptoclub Using Mathematics to Make and Break Secret Codes Janet Beissinger Vera Pless A K Peters Wellesley, Massachusetts

2 Editorial, Sales, and Customer Service Office A K Peters, Ltd. 888 Worcester Street, Suite 230 Wellesley, MA Copyright 2006 by The Board of Trustees of the University of Illinois. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner, with the exception of photocopying for classroom and educational use only, which is permitted. This material is based upon work supported by the National Science Foundation under Grant No Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The Library of Congress has cataloged the book associated with these materials as follows: Beissinger, Janet. The cryptoclub : using mathematics to make and break secret codes / Janet Beissinger, Vera Pless. p. cm. ISBN-13: (alk. paper) ISBN-10: X (alk. paper) 1. Mathematics--Juvenile literature. 2. Cryptography--Juvenile literature. I. Pless, Vera. II. Title. QA40.5.B dc

3 Contents Unit 1 Introduction to Cryptography Chapter 1 Caesar Ciphers Chapter 2 Sending Messages with Numbers Chapter 3 Breaking Caesar Ciphers Unit 2 Substitution Ciphers Chapter 4 Keyword Ciphers Chapter 5 Letter Frequencies Chapter 6 Breaking Substitution Ciphers W1 W5 W13 W19 W23 W26 Unit 3 Vigenère Ciphers Chapter 7 Combining Caesar Ciphers W29 Chapter 8 Cracking Vigenère Ciphers When You Know the Key Length W35 Chapter 9 Factoring W39 Chapter 10 Using Common Factors to Crack Vigenère Ciphers W49 Unit 4 Modular (Clock) Arithmetic Chapter 11 Introduction to Modular Arithmetic Chapter 12 Applications of Modular Arithmetic W67 W75

4 Unit 5 Multiplicative and Affine Ciphers Chapter 13 Multiplicative Ciphers Chapter 14 Using Inverses to Decrypt Chapter 15 Affine Ciphers Unit 6 Math for Modern Cryptography Chapter 16 Finding Prime Numbers Chapter 17 Raising to Powers Unit 7 Public Key Cryptography Chapter 18 The RSA Cryptosystem Chapter 19 Revisiting Inverses in Modular Arithmetic Chapter 20 Sending RSA Messages W81 W93 W105 W115 W123 W131 W133 W137

5 Chapter 1: Caesar Ciphers (Text page 4) Answer Key Caesar cipher with shift of 3 1. a. Encrypt keep this secret with a shift of 3. b. Encrypt your teacher s name with a shift of 3. Answers will vary. Sample: 2. Decrypt the answers to the following riddles. They were encrypted using a Caesar cipher with a shift of 3. a. Riddle: What do you call a sleeping bull? Answer: b. Riddle: What s the difference between a teacher and a train? Answer: Chapter 1: Caesar Ciphers W1

6 (Text page 5) Caesar cipher with shift of 4 3. Decrypt the following note Evie wrote to Abby. She used a Caesar cipher with a shift of 4 like the one above. 4. Use a shift of 3 or 4 to encrypt someone s name. It could be someone in your class or school or someone your class has learned about. (You ll use this to play Cipher Tag.) Answers will vary. Sample (shift 4): W2 Chapter 1: Caesar Ciphers

7 (Text pages 6 7) 5. a. Encrypt private information using a cipher wheel with a shift of 5. (Shift the inner wheel five letters counterclockwise.) b. Encrypt your school s name using a cipher wheel with a shift of 8. Answers may vary. Sample: Use your cipher wheel to decrypt the answers to the following riddles: 6. Riddle: What do you call a dog at the beach? Answer (shifted 4): 7. Riddle: Three birds were sitting on a fence. A hunter shot one. How many were left? Answer (shifted 8): Chapter 1: Caesar Ciphers W3

8 (Text page 7) 8. Riddle: What animal keeps the best time? Answer (shifted 10): 9. Write your own riddle and encrypt the answer. Put your riddle on the board or on a sheet of paper that can be shared with the class later on. (Tell the shift.) Answers will vary. Check students work. Sample: Riddle: What time is it when an elephant sits on your fence? (shift 10) Answer: W4 Chapter 1: Caesar Ciphers

9 Chapter 2: Sending Messages with Numbers (Text page 10) 1. a. Riddle: What kind of cookies do birds like? Answer: b. Riddle: What always ends everything? Answer: ***Return to Text*** 2. a. Encrypt using the cipher strip at the top of the page. b. Encrypt using this cipher strip that is shifted 3. c. Describe how you can use arithmetic to get your answer to 2b from your answer to 2a. Add 3 to each number. Chapter 2: Sending Messages with Numbers W5

10 (Text page 11) 3. Encrypt the following with the given shift: a. shift 4 b. shift 5 c. shift 3 (What is different about encrypting the letter x?) Adding 3 gives 26, which is ***Return to Text*** 4. What numbers between 0 and 25 are equivalent on the circle to the following numbers? a b c d e f a number that is not on the cipher strip. 5. Describe an arithmetic pattern that tells how to match a number greater than 25 with an equivalent number between 0 and 25. Answers may vary. Sample: Subtract 26 from each number greater than 25. If your answer is still greater than 25, subtract again. Repeat until you get a number less than or equal to Encrypt each word by adding the given amount. Your numbers should end up between 0 and 25. a. add 4 b. add 10 W6 Chapter 2: Sending Messages with Numbers

11 (Text page 12) Cipher strip (no shift) 7. Jenny encrypted this name by adding 3. Decrypt to find the name. 8. Riddle: Why doesn t a bike stand up by itself? Answer (encrypted by adding 3): 9. Riddle: What do you call a monkey who loves to eat potato chips? Answer (encrypted by adding 5): 10. Riddle: What is a witch s favorite subject? Answer (encrypted by adding 7): 11. Challenge. This is a name that was encrypted by adding 3. a. Decrypt by subtracting. b. What happens to the 1? What can you do to fix the problem? Sample 1: Subtracting 3 from 1 gives a negative number. If we wrap the numbers in a circle, then counting back 3 from 1 gives 24. Sample 2: The name must be Timmy, so -2 must correspond to y. Chapter 2: Sending Messages with Numbers W7

12 (Text page 13) 12. What numbers between 0 and 25 are equivalent on the circle to the following numbers? a b c d e f Describe an arithmetic pattern that tells how to match a number less than 0 with an equivalent number between 0 and 25. Answers may vary. Sample: Add 26 to each number less than 0. If your sum is still negative, add 26 again. Repeat until the sum is a number between 0 and Decrypt by subtracting. Replace negative numbers with equivalent numbers between 0 and 25. a. subtract 3 b. subtract 10 c. subtract Riddle: What do you call a chair that plays guitar? Answer (encrypted by adding 10): 16. Riddle: How do you make a witch itch? Answer (encrypted by adding 20): W8 Chapter 2: Sending Messages with Numbers

13 (Text page 16) 17. a. To decrypt the riddle in Question 15, you could subtract 10. What number could you add to get the same answer as subtracting 10? 16 b. Here is the answer to the riddle in Question 15. Decrypt it again, adding or subtracting as necessary to avoid negative numbers and numbers greater than 25. Sample: To decrypt 10, subtract 10: = 0, which corresponds to a. (You could also add 16, but this would involve a number greater than 25.) To decrypt 1, add 16: = 17, which corresponds to r. (You could also subtract 10, but this would involve a negative number.) 18. a. Suppose that you encrypted a message by adding 9. Tell two different ways you could decrypt it. Sample: Subtract 9 or add 17. b. This message was encrypted by adding 9. Decrypt by adding or subtracting to avoid negative numbers and numbers greater than 25. Sample: To decrypt 5, add 17: = 22, which corresponds to w. (You could also subtract 9, but this would involve a negative number.) To decrypt 13, subtract 9: 13 9 = 4, which corresponds to e. (You could also add 17, but this would involve a number greater than 25.) Chapter 2: Sending Messages with Numbers W9

14 (Text page 16) 19. a. Suppose that you encrypted a message by adding 5. Tell two different ways you could decrypt it. Subtract 5 or add 21. b. In general, suppose that you encrypted a message by adding an amount n. Tell two different ways you could decrypt it. Subtract n or add (26 n). For Questions 20 23, add or subtract as necessary to make your calculations simplest. 20. Riddle: Imagine that you re trapped in a haunted house with a ghost chasing you. What should you do? Answer (encrypted by adding 10): 21. Riddle: Why must a doctor control his temper? Answer (encrypted by adding 11): W10 Chapter 2: Sending Messages with Numbers

15 (Text page 17) 22. Riddle: What is the meaning of the word coincide? Answer (encrypted by adding 7): 23. Abby was learning about life on the frontier. Peter, she said, Where is the frontier? Decrypt Peter s reply (encrypted by adding 13). Chapter 2: Sending Messages with Numbers W11

16 Here are some blank tables for you to make your own messages. W12 Chapter 2: Sending Messages with Numbers

17 Chapter 3: Breaking Caesar Ciphers (Text page 21) 1. Decrypt Dan's note to Tim. 2. Decrypt Dan's second note to Tim. Chapter 3: Breaking Caesar Ciphers W13

18 (Text page 22) 3. Decrypt each answer by first figuring out the keys. Let the one-letter words help you. a. Riddle: What do you call a happy Lassie? Answer: b. Riddle: Knock, knock. Who s there? Cash. Cash who? Answer: c. Riddle: What s the noisiest dessert? Answer: 4. Decrypt the following quotation: Albert Einstein W14 Chapter 3: Breaking Caesar Ciphers

19 (Text page 24) Decrypt each of the following quotations. Tell the key used to encrypt Theodore Roosevelt Key = 10 Will Rogers Key = 5 Chapter 3: Breaking Caesar Ciphers W15

20 (Text page 25) Thomas A. Edison Key = 8 Albert Camus Key = 13 W16 Chapter 3: Breaking Caesar Ciphers

21 (Text page 25) 9. Thomas A. Edison Key = Challenge. Thomas A. Edison Key = 10 Chapter 3: Breaking Caesar Ciphers W17

22 You can use this page for your own messages. W18 Chapter 3: Breaking Caesar Ciphers

23 Chapter 4: Keyword Ciphers (Text page 31) Write the keyword ciphers in the tables. Decrypt the answers to the riddles. 1. Keyword: DAN, Key letter: h Riddle: What is worse than biting into an apple and finding a worm? Answer: 2. Keyword: HOUSE, Key letter: m Riddle: Is it hard to spot a leopard? Answer: 3. Keyword: MUSIC, Key letter: d Riddle: What part of your body has the most rhythm? Answer: Chapter 4: Keyword Ciphers W19

24 (Text page 31) 4. Keyword: FISH, Key letter: a Riddle: What does Mother Earth use for fishing? Answer: 5. Keyword: ANIMAL, Key letter: g Riddle: Why was the belt arrested? Answer: 6. Keyword: RABBIT, Key letter: f Riddle: How do rabbits travel? Answer: 7. Keyword: MISSISSIPPI, Key letter: d Riddle: What ears cannot hear? Answer: W20 Chapter 4: Keyword Ciphers

25 (Text page 32) 8. Keyword: SKITRIP, Key letter: p (It is a long message, so you may want to share the work with a group.) Chapter 4: Keyword Ciphers W21

26 (Text page 32) 9. Create your own keyword cipher. Answers will vary. Check students work. Keyword: Key letter: On your own paper, encrypt a message to another group. Tell them your keyword and key letter so they can decrypt. Here are extra tables to use when encrypting and decrypting other messages. W22 Chapter 4: Keyword Ciphers

27 Chapter 5: Letter Frequencies (Text page 37) CLASS ACTIVITY: Finding Relative Frequencies of Letters in English Part 1. Collecting data from a small sample. a. Choose about 100 English letters from a newspaper or other English text. (Note: If you are working without a class, choose a larger sample around 500 letters. Then skip Parts 1 and 2.) b. Work with your group to count the As, Bs, etc., in your sample. c. Enter your data in the table below. Answers will vary. Check student work. Some students will use tally marks. Others will count the letters directly and write only numbers. Letter Frequencies for Your Sample Part 2. Combining data to make a larger sample. a. Record your data from Part 1 on your class s Class Letter Frequencies table. (Your teacher will provide this table on the board, overhead, or chart paper.) Note: This table is on page XX. b. Your teacher will assign your group a few rows to add. Enter your sums in the group table. Chapter 5: Letter Frequencies W23

28 (Text pages 37 38) Part 3. Computing relative frequencies. Enter your class s combined data from the Total for All Groups column of Part 2 into the Frequency column. Then compute the relative frequencies. Answers will depend on your class s data. W24 Chapter 5: Letter Frequencies

29 (Text page 38) 1. a. What percent of the letters in the class sample were the letter T? % Answer to 1a depends on your class s data. b. About how many Ts would you expect in a sample of 100 letters? Answer to 1b is 100 times answer to 1a. c. If your sample was about 100 letters, was your answer to 1b close to the number of Ts you found in your sample? Answers will probably be close 2. a. What percent of the letters in the class sample were the letter E? % Answer to 2a depends on your class s data. b. About how many Es would you expect in a sample size of 100? Answer to 2b is 100 times answer to 2a. c. About how many Es would you expect in a sample of 1000 letters? Answer to 2c is 1000 times answer to 2a. 3. Arrange the letters in your class table in order, from most common to least common. Table depends on your class s data. Sample rows shown: 4. The table on Page 39 of the text shows frequencies of letters in English computed using a sample of about 100,000 letters. How is your class data the same as the data in that table? How is it different? Why might it be different? The general pattern of the tables will probably be similar. The most common letters will probably be E, T, A, O, and I, although not necessarily in that order. Ask whether the data for each group had the same most common letters as the data for the whole class. Discuss the benefits of taking a larger sample. Chapter 5: Letter Frequencies W25

30 Chapter 6: Breaking Substitution Ciphers (Text page 49) 1. Use frequency analysis to decrypt Jenny s message, which is shown on the following page. a. Record the number of occurrences (frequency) of each letter in her message. Then compute the relative frequencies. Some students might use tallies. Others will just record the numbers. Letter Frequencies for Jenny s Message W26 Chapter 6: Breaking Substitution Ciphers

31 (Text page 49) b. Arrange letters in order from the most common to the least common. c. Now decrypt Jenny s message, using the frequencies to help you guess the correct substitutions. Record your substitutions in the Substitution Table below the message. Tip: Use pencil! Substitution Table Jenny s Message Chapter 6: Breaking Substitution Ciphers W27

32 (Text page 49) 2. Here is another message to decrypt using frequency analysis. The relative frequencies have been computed for you. Record your substitutions in the Substitution Table below the message. Tip: Use pencil! Message 2 Substitution Table W28 Chapter 6: Breaking Substitution Ciphers

33 Chapter 7: Combining Caesar Ciphers (Text pages 56 57) 1. Encrypt using a Vigenère cipher with keyword DOG. 2. Encrypt using a Vigenère cipher with keyword CAT. ***Return to Text*** 3. Decrypt using a Vigenère cipher with keyword CAT. 4. Decrypt using a Vigenère cipher with keyword LIE. Mark Twain Chapter 7: Combining Caesar Ciphers W29

34 (Text page 58) 5. Use the Vigenère square not a cipher wheel to finish encrypting: 6. Use the Vigenère square to decrypt the following. (Keyword: BLUE) 7. Use either the cipher-wheel method or the Vigenère-square method to decrypt the following quotations from author Mark Twain. a. Keyword: SELF b. Keyword: READ W30 Chapter 7: Combining Caesar Ciphers

35 (Text page 59) 8. Use either the cipher-wheel method or the Vigenère-square method to decrypt the following quotes from Mark Twain. a. Keyword: CAR b. Keyword: TWAIN c. Keyword: NOT Chapter 7: Combining Caesar Ciphers W31

36 (Text page 60) 9. Use either the cipher-wheel method or the Vigenère-square method to decrypt the following quotations. a. Keyword: WISE Thomas Jefferson b. Keyword: STONE Chinese Proverb W32 Chapter 7: Combining Caesar Ciphers

37 (Text page 60) 10. Find a quote from a famous person. Encrypt it using a Vigenère Cipher. Use it to play Cipher Tag. Chapter 7: Combining Caesar Ciphers W33

38 (Text page 60) 11. Challenge. Explore how to describe a Vigenère cipher using numbers. In Chapter 2, you worked with number messages. You described Caesar ciphers with arithmetic by adding to encrypt and subtracting to decrypt. The Vigenère Cipher can be described with arithmetic too. Instead of writing the keyword repeatedly, change the letters of the keyword to numbers and write the numbers repeatedly. Then add to encrypt. For an example, see page 60 of the text. Encrypt and decrypt your own message with this method. W34 Chapter 7: Combining Caesar Ciphers

39 Chapter 8: Cracking Vigenère Ciphers When You Know the Key Length (Text pages 72 73) CLASS ACTIVITY. Finish Decrypting the Girls Message Finish decrypting the Girls Message (key length 4) on pages W36 W37. Your teacher will assign your group 3 or 4 lines of the message to work with. 1. First wheel. The letters for the first wheel are already decrypted. What letter was matched with a? D 2. Second wheel a. Use the table on page 72 of the text to decide how to turn the second wheel. Then decrypt the letters with 2 underneath in your assigned lines. b. What letter did you match with a? I 3. Third wheel a. Find the number of As, Bs, Cs, etc. among the letters with 3 underneath. Record your data in the tables on page W38. b. Use the class data from 3a to decide how to turn the third wheel. Then decrypt the letters with 3 underneath in your assigned lines. c. What letter did you match with a? M 4. Fourth wheel a. Use the partly decrypted message to guess how to decrypt one of the letters with 4 underneath. Use this to figure out what the fourth wheel must be. Then decrypt the rest of your assigned lines. b. What letter did you match with a? E 5. What was the keyword? D I M E Chapter 8: Cracking Vigenère Ciphers When You Know the Key length W35

40 (Text page 70 73) The Girls Message (continued) W36 Chapter 8: Cracking Vigenère Ciphers When You Know the Key length

41 (Text page 70 73) Chapter 8: Cracking Vigenère Ciphers When You Know the Key length W37

42 (Text page 70 73) Tables for the third wheel of the Girls Message. What line numbers are assigned to your group? Answers will vary. To save work, count the letters in your assigned lines only. Then combine data with your class to get a total. Frequency in Your Assigned Lines Class Total W38 Chapter 8: Cracking Vigenère Ciphers When You Know the Key length

43 Chapter 9: Factoring (Text page 76) 1. Find all factors of the following numbers: a. 15 1, 3, 5, 15 b. 24 1, 2, 3, 4, 6, 8, 12, 24 c. 36 1, 2, 3, 4, 6, 9, 12, 18, 36 d. 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 e. 23 1, List four multiples of 5. Answers should include 4 numbers from this list: 5, 10, 15, 20, 25, List all prime numbers less than 30. 2, 3, 5, 7, 11, 13, 17, 19, 23, List all composite numbers from 30 to , 32, 33, 34, 35, 36, 38, 39, 40 Chapter 9: Factoring W39

44 (Text page 77) 5. Use a factor tree to find the prime factorization of each of the following numbers: a. Factor trees will vary. Sample trees are shown. b. c. a. 24 = b. 56 = c. 90 = W40 Chapter 9: Factoring

45 (Text page 79) 6. Circle the numbers that are divisible by 2. How do you know? a. 284 ends in 4 b. 181 c. 70 ends in 0 d ends in 6 7. Circle the numbers that are divisible by 3. How do you know? a = 18; divisible by 3 b = 10; not divisible by 3 c = 7; not divisible by 3 d = 21; divisible by 3 8. Circle the numbers that are divisible by 4. How do you know? a is divisible by 4 b is divisible by 4 c is not divisible by 4 d is divisible by 4 9. Circle the numbers that are divisible by 5. How do you know? a. 80 ends in 0 b. 995 ends in 5 c. 232 d. 444 Chapter 9: Factoring W41

46 (Text page 79) 10. Circle the numbers that are divisible by 6. How do you know? a. 96 divisible by 2 and 3 (9 + 6 = 15), therefore divisible by 6 b. 367 not divisible by 3 since = 16; also not divisible by 2 c. 642 divisible by 2 and 3 (since = 12) d. 842 not divisible by 3 since = Circle the numbers that are divisible by 9. How do you know? a = 9; divisible by 9 b = 9; divisible by 9 c = 15; not divisible by 9 d = 9; divisible by Circle the numbers that are divisible by 10. How do you know? a. 240 The circled numbers end in 0. b c. 60 d W42 Chapter 9: Factoring

47 (Text page 81) 13. Use a factor tree to find the prime factorization of each of the following numbers. Write each factorization using exponents. a. Factor trees will vary. Sample trees are shown. b. a = b = Chapter 9: Factoring W43

48 (Text page 81) 13. c. d. c. 357 = d. 56,133 = W44 Chapter 9: Factoring

49 (Text page 81) 13. e. f. e. 14,625 = f = Chapter 9: Factoring W45

50 (Text page 82) 14. Find the common factors of the following pairs of numbers: a. 10 and 25 Common factors: 1, 5 10 = 2 5 factors: 1, 2, 5, = 5 5 factors: 1, 5, 25 b. 12 and 18 Common factors: 1, 2, 3, 6 12 = factors: 1, 2, 3, 4, 6, = factors: 1, 2, 3, 6, 9, 18 c. 45 and 60 Common factors: 1, 3, 5, = Instead of listing all factors, it is quicker to 60 = combine common prime factors to get common factors: 1, 3, 5, Find the greatest common factor of each of the following pairs of numbers: a. 12 and 20 Greatest common factor: 4 12 = = b. 50 and 75 Greatest common factor: = = c. 30 and 45 Greatest common factor: = = W46 Chapter 9: Factoring

51 (Text page 82) 16. For each list of numbers, factor the numbers into primes and then find all common factors for the list. Use the space beside the problems for any factor trees you want to make. a. 14 = = = 2 5 Common factor(s): 1, 2 b. 66 = = = Common factor(s): 1, 2, 3, 2 3 = 6 c. 30 = = = Common factor(s): combine 2, 3, and 5: 1, 2, 3, 5, 2 3 = 6, 2 5 = 10, 3 5 = 15, = 30 Chapter 9: Factoring W47

52 Continue to the next chapter. W48 Chapter 9: Factoring

53 Chapter 10: Using Common Factors to Crack Vigenère Ciphers (Text page 88) These problems involve entries Meriwether Lewis wrote in his journal during the Lewis and Clark Expedition. (You might notice that the spelling is not always the same as modern-day spelling, but we show it as it originally was written.) 1. Sunday, May 20, 1804 a. Circle all occurrences of the in the message above. Include examples such as there in which the occurs as part of a word. b. Find the distance from the beginning of the last the in the 5 th line to the beginning of the in the 6 th line. 30 c. Choose a keyword from RED, BLUE, ARTICHOKES, TOMATOES that will encrypt in exactly in the same way the two occurrences of the in the following phrase (from the last sentence of the message). Then, use it to encrypt the phrase. Answers will vary. Could be RED or ARTICHOKES, since their lengths (3 and 10) are factors of 30. See next page for encryption with keyword RED. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W49

54 (Text page 88) 1. d. Choose a keyword from RED, BLUE, ARTICHOKES, TOMATOES that will encrypt in different ways the two occurrences of the in the phrase. Then use it to encrypt. Answers will vary. Could be BLUE or TOMATOES since their lengths are not factors of 30. See below for encryption with keyword TOMATOES. e. Of the keywords you have not used, which would encrypt the two occurrences of the in the phrase above in the same way? RED and ARTICHOKE (Student should use the word not used for 8c.) in different ways? BLUE or TOMATOES (whichever was not used for 8d) Give reasons for your answers: Sample: RED and ARTICHOKE could encrypt the in the same way since their lengths (3 and 10) are factors of 30, which is the distance between occurrences of the. Here are encryptions of the same phrase with different keywords: RED: TOMATOES: KLH IELE EQU NRZRHU GDGX FCEUB JRLRG GEUKC MVQ RTWR SGR VOBBIV VOBT VZEJD TAUGR XZX DMRMM W50 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

55 (Text page 89) 2. Wednesday, April 7, 1805 a. Circle the occurrences of the in the above message. b. Find the distance from the beginning of the in the second line to the beginning of the in the third line. 36 List all keyword lengths that would cause these words to be encrypted the same way. 2, 3, 4, 6, 9, 12, 18 c. Find the distance from the beginning of the in the third line to the beginning of these in the fourth line. 63 List all keyword lengths that would cause the in these strings to be encrypted the same way. 3, 7, 9, 21 d. What keyword length(s) would cause all three occurrences of the described in 2b and 2c to be encrypted the same way? 3 and 9 (common factors of 36 and 63) Chapter 10: Using Common Factors to Crack Vigenère Ciphers W51

56 (Text page 89) 2. e. Choose the keyword from the following list that will cause all three occurrences of the described in 2b and 2c to be encrypted the same way. PEAR, APPLE, CARROT, LETTUCE, CUCUMBER, ASPARAGUS, WATERMELON, CAULIFLOWER f. Write your chosen keyword above the message below. Encrypt each occurrence of the. (You don t have to encrypt the entire message.) Note: We provide the complete encryption here, but students are only asked to encrypt occurrences of the to show they are encrypted the same. W52 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

57 Continue to the next page for Problem 3. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W53

58 (Text page 90) 3. a. Underline strings of letters that repeat in the girls message below: Some of the repeated strings are shown below. W54 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

59 (Text page 90) 3. b. Complete the table below. Include the strings shown in the table as well as the strings you found in 3a. Combine student answers to get a class table. Sample. See below for more repeated strings. c. Is the key length always, usually, or sometimes a factor of the distance between strings? sometimes Distance for repeated strings not listed above: VIVAQ 212, MRG 68 and 164; QVH 88; FRG 160; NHAEI 184; FLO 132; NSBA 232; XKMK 204 and 120; PMPM 188; BASN 100; BTI 28, 67, and 96; MJWMD 52. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W55

60 (Text page 92) 4. a. Underline or circle strings that repeat in Grandfather s message below. Include at least two strings whose distances aren t in the table in 4b. Repetitions of GZS are already underlined. W56 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

61 (Text page 92) 4. b. Find the distances between the occurrences you found and record them in the table. Then, for each distance in the table, tell whether 3 is a factor. Repeated Strings in Grandfather s Message c. How did you determine whether 3 was a factor of a number? I added the digits. Three is a factor of a number if the sum of its digits is a multiple of 3. d. Do you think 3 is a good guess for the key length of Grandfather s message? Why or why not? Yes. Three is a factor of most distances. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W57

62 (Text page 93) 5. Decrypt Grandfather s message. To save time, use the information in the table to help choose how to turn each Caesar wheel. (It is a long message so you might want to share the work.) Most Common Letters for Each Wheel Keyword: S O N Grandfather s Message (continued) W58 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

63 (Text page 93) Chapter 10: Using Common Factors to Crack Vigenère Ciphers W59

64 (Text page 94) 6. Below is Grandfather s message encrypted with a different keyword. The goal is to find the key length. (You don t have to decrypt the message since you already know it.) a. Underline at least three pairs of repeated strings, including at least two whose distances are not in the table. Then find the distances between occurrences of those strings. b. Factor the distances given in the table. c. Make a reasonable guess about what the length of the keyword might be. 4 Explain why your answer is reasonable. 2 2 = 4 is a factor of every distance. (The keyword used was GOLD.) W60 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

65 (Text pages 94 95) 7. Grandfather s message is encrypted again with a different keyword. (You don t have to decrypt.) a. Underline at least three pairs of repeated strings, including at least two whose distances are not in the table. Then find the distances between occurrences of those strings. b. Factor the distances given in the table. \ c. Make a reasonable guess about what the length of the keyword might be. 6 Explain why your answer is reasonable. 2 3 = 6 is a factor of all numbers. (The keyword used was SECRET) Chapter 10: Using Common Factors to Crack Vigenère Ciphers W61

66 (Text pages 94 95) 8. Here is Grandfather s message again, encrypted with a different keyword. a. Underline at least three pairs of repeated strings, including at least two whose distances are not in the table. Then find the distances between occurrences of those strings. b. Factor the distances given in the table. c. Make a reasonable guess about what the length of the keyword might be. 5 Explain why your answer is reasonable. 5 is a factor of most of the distances. 3 would also be a reasonable guess. (The keyword used was APPLE) W62 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

67 Continue to the next page for Problem 9. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W63

68 (Text page 96) 9. The following message is encrypted with a Vigenère cipher. Collect data to guess the key, then crack the message. Use the suggestions in 9a e on the following pages to share the work with your class. W64 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

69 (Text page 96) 9. a. Find strings that repeat. Then find and factor the distances between them. Share your data with your class. b. What is a likely key length? 4 Write numbers under the message to show letters for each wheel. c. Your teacher will divide the class into groups and assign your group one of the wheels. Find the frequency of the letters for your wheel. See p. W65a W65a W65b for data for from Wheels 2, 3, 3, and 4. Chapter 10: Using Common Factors to Crack Vigenère Ciphers W65

70 W65a Chapter 10: Using Common Factors to Crack Vigenère Ciphers

71 Chapter 10: Using Common Factors to Crack Vigenère Ciphers W65b

72 (Text page 96) 9. d. Record class data about the most common letters for each wheel. e. Your teacher will assign your group a few lines of the message to decrypt. Share your decrypted lines with the class. What keyword did you use? gold 10. Describe in your own words how to crack a Vigenère cipher when you do not know anything about the keyword. To break a Vigenère cipher when you do not know the key, a. Make a good guess about what the key length is. Do this by looking for strings of letters that repeat and finding the distance between repeated occurrences. A good guess for the key length is a number that is a common multiple of most of the distances. b. If the key length is known (or suspected), divide the message into as many groups as there are letters in the keyword. The first letter goes in the first group, the second letter goes in the second group, etc. Then decrypt each group separately as you would a Caesar cipher. Finding the most common letters in each group helps. W66 Chapter 10: Using Common Factors to Crack Vigenère Ciphers

73 Chapter 11: Introduction to Modular Arithmetic (Text page 104) 1. Lilah had a play rehearsal that started at 11:00 on Saturday morning. The rehearsal lasted 3 hours. What time did it end? 2 PM 2. Peter was traveling with his family to visit their grandmother and their cousins, Marla and Bethany, near Pittsburgh. The car trip would take 13 hours. If they left at 8:00 AM, what time would they arrive in Pittsburgh? 9 PM 3. The trip to visit their other grandmother takes much longer. First they drive for 12 hours, then stop at a hotel and sleep for about 8 hours. Then they drive about 13 hours more. If they leave at 10:00 AM on Saturday, when will they get to their grandmother s house? 7 PM Monday 4. Use clock arithmetic to solve the following: a = 3 b = 7 c = 10 d = 1 5. Jenny s family is planning a 5-hour car trip. They want to arrive at 2 PM. What time should they leave? 9 AM 6. In Problem 5, we moved backward around the clock. This is the same as subtracting in clock arithmetic. Solve the following subtraction problems using clock arithmetic. Use the clock, if you like, to help you: a. 3 7 = 8 b. 5 6 = 11 c. 2 3 = 11 d = 7 Chapter 11: Introduction to Modular Arithmetic W67

74 (Text page 106) 7. Write the following 12-hour times using the 24-hour system: a. 3 PM 15:00 b. 9 AM 9:00 c. 11:15 PM 23:15 d. 4:30 AM 4:30 e. 6:45 PM 18:45 f. 8:30 PM 20:30 8. Write the following 24-hour times as 12-hour times, using AM or PM. a. 13:00 1:00 PM b. 5:00 5:00 AM c. 19:15 7:15 PM d. 21:00 9:00 PM e. 11:45 11:45 AM f. 15:30 3:30 PM 9. Use clock arithmetic on a 24-hour clock to solve the following: a = 2 b = 4 c = 2 d = 20 W68 Chapter 11: Introduction to Modular Arithmetic

75 (Text page 106) 10. Solve the following on a 10-hour clock: a = 2 b = 3 c = 4 d = 5 e. 6 8 = 8 f = Challenge: Is always 4? Find a clock for which this is not true. Answers may vary. Sample: No; On a 3-hour clock, = 1. Chapter 11: Introduction to Modular Arithmetic W69

76 (Text page 107) 12. The figure shows numbers wrapped around a 12-hour clock. a. List all numbers between 1 and 48 that have the same position on a 12-hour clock as 3. 3, 15, 27, 39 b. If the number wrapping continues, what numbers between 49 and 72 would have the same position on a 12-hour clock as 3? 51, a. List all numbers between 1 and 48 that have the same position on a 12-hour clock as 8. 8, 20, 32, 44 b. If the number wrapping continues, what numbers between 49 and 72 would have the same position on a 12-hour clock as 8? 56, a. How can you use arithmetic to describe numbers that have the same position on a 12-hour clock as 5? Answers may vary. Sample: Add 12 successively. b. What numbers between 49 and 72 have the same position on the 12-hour clock as 5? 53, 65 W70 Chapter 11: Introduction to Modular Arithmetic

77 (Text page 108) 15. List three numbers equivalent to each number. Answers may vary. Samples are given. a. 6 mod 12 18, 30, 42 b. 9 mod 12 21, 33, List three numbers equivalent to each number. Answers may vary. Samples are given. a. 2 mod 10 12, 22, 32 b. 9 mod 10 19, 29, 39 c. 0 mod 10 10, 20, List three numbers equivalent to each number. Answers may vary. Samples are given. a. 1 mod 5 6, 11, 16 b. 3 mod 5 8, 13, 18 c. 2 mod 5 7, 12, 17 Chapter 11: Introduction to Modular Arithmetic W71

78 (Text page 110) 18. Reduce each number. a. 8 mod 5 = 3 b. 13 mod 5 = 3 c. 6 mod 5 = 1 d. 4 mod 5 = Reduce each number. a. 18 mod 12 = 6 c. 36 mod 12 = Reduce each number. a. 8 mod 3 = 2 c. 16 mod 11 = 5 b. 26 mod 12 = 8 d. 8 mod 12 = 8 b. 13 mod 6 = 1 d. 22 mod 7 = 1 W72 Chapter 11: Introduction to Modular Arithmetic

79 (Text page 110) 21. Reduce each number. a. 4 mod 12 = 8 b. 1 mod 12 = 11 c. 6 mod 12 = 6 d. 2 mod 12 = Reduce each number. a. 4 mod 10 = 6 b. 1 mod 10 = 9 c. 6 mod 10 = 4 d. 2 mod 10 = Reduce each number. a. 3 mod 5 = 2 b. 1 mod 5 = 4 c. 8 mod 5 = 3 d. 7 mod 5 = Reduce each number. a. 2 mod 24 = 22 b. 23 mod 20 = 3 c. 16 mod 11 = 5 d. 3 mod 20 = 17 Chapter 11: Introduction to Modular Arithmetic W73

80 Continue to the next chapter. W74 Chapter 11: Introduction to Modular Arithmetic

81 Chapter 12: More Modular Arithmetic (Text pages ) 1. Reduce the following numbers mod 26: a. 29 = 3 mod 26 c. 12 = 12 mod 26 e. 4 = 22 mod 26 g. 10 = 16 mod 26 b. 33 = 7 mod 26 d. 40 = 14 mod 26 f. 52 = 0 mod 26 h. 7 = 19 mod Encrypt the name Jack using the times-5 cipher. The first two letters are done for you. The rule for encrypting is given in the table. 3. Encrypt cryptography using a times-3 cipher. The first two letters are done for you. Chapter 12: More Modular Arithmetic W75

82 (Text pages ) 4. Reduce each number. a. 175 mod 26 = 19 b. 106 mod 26 = 2 c. 78 mod 26 = 0 d. 150 mod 26 = Reduce each number. (Hint: Try subtracting multiples of 26 such as = 260.) a. 586 mod 26 = 14 c. 541 mod 26 = 21 b. 792 mod 26 = 12 d. 364 mod 26 = 0 ***Return to Text*** 6. Use a calculator to help you reduce each number. a. 254 mod 24 = 14 c. 827 mod 26 = 21 b. 500 mod 5 = 0 d mod 26 = 18 e. 700 mod 9 = 7 f. 120 mod 11 = 10 W76 Chapter 12: More Modular Arithmetic

83 (Text pages ) 7. Reduce each number. a. 500 mod 7 = 3 b mod 24 = 16 c. 25,000 mod 5280 = 3880 d. 10,000 mod 365 = Choose one of the numbers you reduced in Problem 6. Write how you would explain to a friend the way you reduced your number. Answers will vary. Here are two sample answers for 827 mod 26 (Problem 6c). Sample 1: I used my calculator to divide = I subtracted 31 to get the decimal remainder, I multiplied this by 26 and got 21. Sample 2: I subtracted multiples of 26 from 827; = 567; = 307; = 47; = 21. ***Return to Text*** 9. Encrypt trick, using a times-11 cipher. Use Tim s shortcut when it makes your work easier. Note: Students may have different opinions about when Tim s shortcut makes their work easier. For example, some might replace 25 with 1, but leave 18, 19, and 24 as is. Chapter 12: More Modular Arithmetic W77

84 (Text page 120) 10. Astronauts left on a Sunday for a mission into space. On what day of the week would they return if they were gone for a. 4 days? Thursday b. 15 days? Monday (since 15 mod 7 = 1) c. 100 days? Tuesday (since 100 mod 7 = 2) d days? Saturday (since 1000 mod 7 = 6) 11. If today is Wednesday, what day of the week will it be in a. 3 days? Saturday b. 75 days? Monday (75 mod 7 = 5; 5 days after Wednesday is Monday) c. 300 days? Tuesday (300 mod 7 = 6; 6 days after Wednesday is Tuesday) Leap Years. There are 365 days in a year, except for leap years. In a leap year, an extra day (February 29) is added, making 366 days. Leap years occur in years divisible by 4, except at the beginning of some centuries. Years that begin new centuries are not leap years unless they are divisible by 400. So 1900 was not a leap year but 2000 was. 12. a was a leap year. What are the next two leap years? 2008, 2012 b. Which of the following century years are leap years? , 2100, 2400 c. Which of the following years were leap years? 1996, , 1776, 1890 W78 Chapter 12: More Modular Arithmetic

85 (Text page 120) 13. If the Fourth of July is on Tuesday this year, on what day of the week will it be next year? (Assume that next year is not a leap year.) Explain how you got your answer. Wednesday. 365 mod 7 = 1; one day after Tuesday is Wednesday. 14. a. What is today s day and date? Sample: Monday, April 3, 2006 b. What day of the week will it be on today s date next year? Your answer will depend on whether or not a leap year is involved. Explain how you got your answer. Sample 1: If today is Monday, April 3, 2006, next year is 2007, which is not a leap year. Therefore, there are 365 days until today s date next year. Since 365 mod 7 = 1, today s date will fall on Tuesday. Sample 2: If today is Tuesday, April 3, 2007, then next year is 2008, which is a leap year. There are 366 days until today s date next year. Since 366 mod 7 = 2, today s date will fall on Thursday. Sample 3: If today is Thursday, April 3, 2008, today s date will fall on Friday next year. Caution: Students might think there will be 366 days before this date next year, because 2008 is a leap year. But February 29, 2008 has already passed. The next February before this date next year is February, Chapter 12: More Modular Arithmetic W79

86 (Text page 120) 15. a. On what day and date will your next birthday be? (You may use a calendar.) Sample: Wednesday, September 13, 2006 b. On what day of the week will your twenty-first birthday be? Answer without using a calendar. Don t forget about leap years. Explain how you got your answer. I will be 12 on September 13, I will be 21 nine years later. My 21 st birthday will be September 13, If there were 365 days in every year, I would be 21 in days after my next birthday. But 2008 and 2012 have an extra day each. Therefore, I will be 21 in = 3287 days mod 7 = 4; 4 days after Wednesday is Sunday. I will be 21 on Sunday, September 13, W80 Chapter 12: More Modular Arithmetic

87 Chapter 13: Multiplicative Ciphers (Text page 126) 1. a. Complete the times-3 cipher table. (Tip: You can use patterns such as multiplying by 3s to multiply quickly.) b. Decrypt the following message Evie wrote using the times-3 cipher. c. Riddle: What has one foot on each end and one foot in the middle? (It was encrypted using the times-3 cipher.) Answer: Chapter 13: Multiplicative Ciphers W81

88 (Text page 126) 2. a. Complete the times-2 cipher table. b. Use the times-2 cipher to encrypt the words ant and nag. Is there anything unusual about your answers? They are both encrypted the same way. c. Make a list of pairs of letters that are encrypted the same way using a times-2 cipher. For example, a and n are both encrypted as A, b and o are both encrypted as C. Writing the first half of the alphabet on top of the second half is one way to list such pairs: d. Make a list of several pairs of words that are encrypted the same way using the times-2 cipher. Answers will vary; examples include ant and nag, as in 2b. e. Decrypt KOI in more than one way to get different English words. Answers will vary. Sample: she & fur f. Does multiplying by 2 give a good cipher? Why or why not? No. Different letters are encrypted the same way. W82 Chapter 13: Multiplicative Ciphers

89 (Text pages ) 3. a. Complete the times-5 cipher table. Then decrypt the quotations. b. Abraham Lincoln c. Albert Einstein Chapter 13: Multiplicative Ciphers W83

90 (Text page 127) 4. a. Complete the times-13 cipher table. b. Encrypt using the times-13 cipher: c. Does multiplying by 13 give a good cipher? Why or why not? No. Different letters are encrypted the same way. It is impossible to decrypt. W84 Chapter 13: Multiplicative Ciphers

91 Class Activity (Text page 127) a. Choose one even number and one odd number between 4 and 25 to investigate. One number should be big, the other small. (Groups that finish early can work on the numbers not yet chosen.) write chosen numbers here sample: 4 15 b. Compute cipher tables using your numbers as multiplicative keys. Decide which of your numbers make good multiplicative keys (that is, which numbers encrypt every letter differently). (Use the extra tables on the back of this page if you compute more than two ciphers.) Sample: Times- 4 Cipher Good key or bad key? bad Sample: Times- 15 Cipher Good key or bad key? good c. Pool your information with the rest of the class. Describe a pattern that tells which numbers give good keys. All odd numbers except 13 give different products and are therefore good keys. (Multiplying by 1 doesn t give a very good cipher since it doesn t change anything, but we do count it as a good key because it encrypts each number differently.) Chapter 13: Multiplicative Ciphers W85

92 Extra Tables: Sample: Times- 6 Cipher Good key or bad key? bad Sample: Times- 17 Cipher Good key or bad key? good Sample: Times- 19 Cipher Good key or bad key? good W86 Chapter 13: Multiplicative Ciphers

93 (Text pages ) 5. Which of the following pairs of numbers are relatively prime? a. 3 and 12 no b. 13 and 26 no (3 is a common factor) (13 is a common factor) c. 10 and 21 rel. prime d. 15 and 22 rel. prime e. 8 and 20 no f. 2 and 14 no (2 is a common factor) (2 is a common factor) 6. a. List 3 numbers that are relatively prime to 26. Answers will vary. Could be any 3 odd numbers except 13. b. List 3 numbers that are relatively prime to 24. Answers will vary. Could be any three numbers that do not have 2 or 3 as prime factors. 7. Which numbers make good multiplicative keys for each of the following alphabets? a. Russian; 33 letters 33 = 3 11, so numbers relatively prime to 33 do not have 3 or 11 as factors. 1, 2, 4, 5, 7, 8, 10, 11. b. Lilah s alphabet, which consists of the 26 English letters and the period, comma, question mark, and blank space 30 = Numbers relatively prime to 30 are 1, 7, 11, 13, 17, 19, 23, 29 c. Korean; 24 letters 24 = 2 3 3; numbers relatively prime to 24 are 1, 5, 7, 11, 13, 17, 19, 23 d. Arabic; 28 letters. This alphabet is used to write about 100 languages, including Arabic, Kurdish, Persian, and Urdu (the main language of Pakistan). 28 = 2 2 7; Numbers relatively prime to 28 are the odd numbers except 7 and 21. e. Portuguese; 23 letters 23 is prime; All numbers from 1 to 22 are good keys since they are all relatively prime to 23. Chapter 13: Multiplicative Ciphers W87

94 (Text page 130) 8. Compute the table for each cipher, then decrypt the quote: a. Times-7 cipher H. Jackson Brown, Jr. W88 Chapter 13: Multiplicative Ciphers

95 (Text page 130) 8. b. Times-9 cipher Anne Morrow Lindbergh Chapter 13: Multiplicative Ciphers W89

96 (Text page 130) 8. c. Times-11 cipher William Shakespeare W90 Chapter 13: Multiplicative Ciphers

97 (Text page 130) 8. d. Times-25 cipher (Hint: 25 1 (mod 26).) Plato Chapter 13: Multiplicative Ciphers W91

98 (Text page 131) 9. Look at your cipher tables from Problem 8. a. How was a encrypted? A Will this be the same in all multiplicative ciphers? Give a reason for your answer. Yes. a corresponds to zero. Zero times any number is always zero, so a will always correspond to A. b. How was n encrypted? N Challenge. Show that this will be the same in all multiplicative ciphers. Hint: Since all multiplicative keys are odd numbers, every key can be written as an even number plus 1. Since the key must be odd, it can be written as key = 2 m + 1 for some m. Then 13 key = 13 (2 m + 1) = 13 2 m + 13 (distributive law) = 26 m (mod 26) 13 (mod 26) This is N. No matter what (odd) key we use, we always get N. W92 Chapter 13: Multiplicative Ciphers

99 Chapter 14: Using Inverses to Decrypt (Text pages ) 1. Compute the following in regular arithmetic. a = 1 b = 1 c = 1 2. Complete: a. b. c. (fill in the box) d. (fill in the box) ***Return to Text*** 3. Test Abby s theory that if you multiply by 3 and then by 9 (and reduce mod 26) you get back what you started with: a. (mod 26) b. (mo (mod 26) c. (mod 26) Chapter 14: Using Inverses to Decrypt W93

100 (Text page 136) 4. Where was Tim s second treasure hidden? Finish decryping his clue to find out. (He used a multiplicative cipher with key 3). W94 Chapter 14: Using Inverses to Decrypt

101 (Text page 137) 5. Look at the tables of multiplicative ciphers you have already worked out. Find the column that has 1 in the product row and use this to find other pairs of numbers that are inverses mod 26. Save these for later. Answers will vary. Students should look back at their tables from Chapter 13. They should list some of the following pairs: 3, 9 5, 21 7, 15 11, 19 17, 23 25, 25 ***Return to Text*** 6. Since 5 21 = (mod 26), 5 and 21 are inverses of each other (mod 26). Find another way to factor 105. Use this to find another pair of mod 26 inverses. 7 and The following was encrypted by multiplying by 21. Decrypt. (Hint: See Problem 6 for the inverse of 21.) Orison Swett Marden Chapter 14: Using Inverses to Decrypt W95

102 (Text page 138) 8. Find another inverse pair by looking at the negatives of the inverses we have already found. Answers may vary. Sample: Begin with 7 and 15, which are inverses (mod 26) and (mod 26) 19 and 11 are inverses, since ( 7) ( 15) (mod 26) (mod 26) 1 (mod 26). 9. What is the inverse of 25 mod 26? (Hint: 25 1 (mod 26).) ( 1) ( 1) (mod 26) 1 (mod 26) Therefore, 25 is its own inverse. W96 Chapter 14: Using Inverses to Decrypt

103 (Text page 138) 10. a. Make a list of all the pairs of inverses you and your classmates have found. (Keep this list to help you decrypt messages.) 3, 9 5, 21 7, 15 11, 19 17, 23 25, 25 b. What numbers are not on your list? (Not all numbers have inverses mod 26.) even numbers and 13 c. Describe a pattern that tells which numbers between 1 and 25 have inverses mod 26. numbers relatively prime to Challenge. Explain why even numbers do not have inverses mod 26. One way to know this is to look at the times-2 cipher table in Chapter 13 it has no 1 in the product row. Another way is to note that an even number times any number is always even it can never be 1 (mod 26). (All numbers equivalent to 1 mod 26 are odd because they are of the form 26 n + 1, which is odd.) Chapter 14: Using Inverses to Decrypt W97

104 (Text page 139) Solve these problems by multiplying by the inverse. 12. Riddle: What word is pronounced wrong by the best of scholars? Answer (encrypted with a times-9 cipher): 13. Riddle: What s the best way to catch a squirrel? Answer (encrypted with a times-15 cipher): W98 Chapter 14: Using Inverses to Decrypt

105 (Text page 139) 14. Challenge. Investigate inverses for one of the alphabets listed below. Find all pairs of numbers that are inverses of each other. a. Russian; 33 letters b. The English alphabet, and the period, comma, question mark, and blank space; 30 letters c. Korean; 24 letters (Note: There is something unusual about the inverses for this alphabet.) a. Russian: 33 letters Numbers relatively prime to 33 have inverses. Since 33 = 3 11, numbers with inverses are 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32. Numbers congruent to 1 mod 33 are: 34, 67, 100, 133, 166, 199, Look for factor pairs of these numbers. 34 = = = 5 20 = = 7 19 Negatives of inverse pairs: ( 2) 31 (mod 33); (mod 33) (mod 33); (mod 33) 5 28 (mod 33); mod mod 33; 25 8 mod (mod 33); (mod 33) 32 is its own inverse, since ( 1) ( 1) (mod 33) 1 (mod 33) Inverses so 2 and 17 are inverses so 10 is its own inverse so 5 and 20 are inverses so 4 and 25 are inverses so 7 and 19 are inverses More Inverses 31, 16 23, 23 28, 13 29, 8 26, 14 32, 32 Chapter 14: Using Inverses to Decrypt W99

106 14. b. The English alphabet, with characters making 30 letters Numbers relatively prime to 30 have inverses. Since 30 = 2 3 5, these are numbers that do not have 2, 3, or 5 as factors. They are 7, 11, 13, 17, 19, 23, 29. Numbers congruent to 1 mod 30 are 31, 61, 91, 121, 151, 181, Look for factor pairs of these numbers. 31 prime 61 prime 91 = 7 13 so 7 and 13 are inverses 121 = so 11 is its own inverse Negatives: 7 23 (mod 30); (mod 30) So 23 and 27 are inverses (mod 30), so (mod 30) 29 is its own inverse. Inverse pairs: 7, 13 11, 11 23, 17 29, 29 Since we have found inverses of all numbers relatively prime to 30 except 19, we suspect 19 is its own inverse. To confirm this, compute = (mod 30) W99a Chapter 14: Using Inverses to Decrypt

107 14. c. Korean; 24 letters Numbers relatively prime to 24 have inverses. Since 24 = , these are numbers that are not multiples of 2 or 3. These are 5, 7, 11, 13, 17, 19, 23. Numbers congruent to 1 mod 24 are 25, 49, 73, 97, 121, Look for factor pairs: Each number is its 25 = 5 5 own inverse. 49 = = Negatives: 5 19 (mod 24) 7 17 (mod 24) (mod 24) +23 1, so 23 is its own inverse. Chapter 14: Using Inverses to Decrypt W99b

108 (Text page 141) 15. Where did Evie s note say to meet? Finish decrypting to find out. Show your work below the message. The calculations in the text show that the encryption key is 5. That means we decrypt by multiplying by the inverse of 5, which is 21. D is shown to be l in the text. Other letters are computed here: O corresponds to = (mod 26) This is i. F corresponds to = (mod 26) This is b. H corresponds to = (mod 26) This is r. Q corresponds to = (mod 26) This is y. W100 Chapter 14: Using Inverses to Decrypt

109 (Text page 141) 16. The following messages were encrypted with multiplicative ciphers. A few letters in each message have been decrypted. For each message, write an equivalence that involves the key. Then solve the equivalence to find the key. Use the inverse of the key to help decrypt. Show your work below the messages. (Note: No table is given, so you decide how you want to organize your work.) a. Ralph Waldo Emerson Plaintext t corresponds to ciphertext P. So, 19 corresponds to key 19 (mod 26) Inverse of 19 is key (19 11) (mod 26) 165 key (mod 26) 9 key (mod 26) To decrypt message, multiply by inverse of 9, which is 3. Here are the first few calculations: Q corresponds to = (mod 26). This is w. X corresponds to = (mod 26). This is r. Chapter 14: Using Inverses to Decrypt W101

110 (Text page 141) 16. b. Mark Twain Key is 15. Decrypt by multiplying by inverse of key: 7. W102 Chapter 14: Using Inverses to Decrypt

111 (Text page 142) 17. For each of the following, find the most common letters in the message. Use this information or other reasoning to guess a few letters of the message. Then find the encryption key by solving an equivalence. Use the inverse of the key to help decrypt. Show your work below the messages. a. Winston Churchill Most common letters in the message are C, O, V, Y. If C is e, then the word CN would begin with e unlikely in English so e is not a good choice for C. So try matching e with O. The equation that results is 14 key 4 (mod 26). But 4 does not have an inverse, so make another match. V could be t since then VFO could be the. The equation that results is 21 key 19 (mod 26). Multiply by the inverse of 19, which is 11: key (19 11) (mod 26). 231 key (mod 26) 23 key (mod 26) The encryption key is 23. So, multiply by its inverse, 17, to decrypt: Sample: H corresponds to 7; 17 7 = (mod 26). This is p. Chapter 14: Using Inverses to Decrypt W103

112 (Text page 142) 17. b. Ralph Waldo Emerson Most common letters in ciphertext: A, G, R, O, X. Key is 21. Decrypt by multiplying by 5. W104 Chapter 14: Using Inverses to Decrypt

113 Chapter 15: Affine Ciphers (Text page 145) 1. How many different additive ciphers are possible? That is, how many different numbers can be keys for additive ciphers? Explain how you got your answer. 25 or 26, depending on whether you count zero. Adding zero doesn t give a very good cipher. 2. How many different multiplicative ciphers are possible? That is, how many different numbers make good keys for multiplicative ciphers? Explain how you got your answer. (Remember, the good multiplicative keys are those that are relatively prime to 26.) (11 or 12, depending on whether you include 1.) The good keys are the numbers relatively prime to 26. These are the odd numbers except 13: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25. We count 1 as a good key because it encrypts each letter differently, but it doesn t really give a very good cipher, since it doesn t change the letters. Chapter 15: Affine Ciphers W105

114 (Text page 146) 3. Encrypt secret using the (3, 7)-affine cipher. 4. Encrypt secret using the (5, 8)-affine cipher. W106 Chapter 15: Affine Ciphers

115 (Text pages ) 5. Some affine ciphers are the same as other ciphers we have already explored. a. What other cipher is the same as the (3, 0)-affine cipher? times-3 cipher b. What other cipher is the same as the (1, 8)-affine cipher? Caesar cipher with shift 8 6. Suppose that Dan and Tim changed their key to get a different affine cipher each day. Would they have enough ciphers to have one for each day of the year? Explain. No. To count the different affine ciphers, count the possible keys (m, b). There are 12 possible good numbers for m and 26 possible numbers for b. Each m can be paired with a different b, so there are = 312 different keys of the form (m, b). If you don t count (1, 0), which wouldn t make a good cipher since it wouldn t change anything, then there are 311 ciphers. Either way you count, there are not enough ciphers for a whole year. ***Return to Text*** 7. Riddle: What insects are found in clocks? Answer (encrypted with a (3, 7)-affine cipher): Chapter 15: Affine Ciphers W107

116 (Text page 147) 8. Decrypt the girls invitation. It was encrypted with a (5, 2)-affine cipher. There is no table given, so you can decide how you would like to organize your work. To decrypt: Subtract 2, then multiply by the inverse of 5, which is 21. Remind students that they don t need to do calculations for the same letters more than once. W108 Chapter 15: Affine Ciphers

117 (Text page 150) 9. Each of the following was encrypted with an affine cipher. A few letters have been decrypted. For each message, write equivalences involving the encryption key (m, b). Solve the equivalences to find the m and b. Then decrypt the message. a. Mahatma Gandhi e matches M, so 4 is encrypted as 12: 12 m 4 + b (mod 26) Also, n matches P, so 13 is encrypted as 15: 15 m 13 + b (mod 26) Subtract equivalences: 15 13m + b (mod 26) ( 12 4m + b (mod 26)) 3 9m + 0 (mod 26) To solve, multiply both sides by the inverse of 9, which is 3: m (mod 26) 1 9 m (mod 26) Substitute 9 = m into one of the equivalences: b (mod 26) b (mod 26) 24 b (mod 26) 2 b (mod 26) The key is (9, 2). To decrypt, subtract 2 and multiply each letter by 3 (the inverse of 9). Samples: C matches 2; 2 2 = 0; 0 3 = 0. This is a. Z matches 25; 25 2 = 23; 23 3 = (mod 26). This is r. R matches 17; 17 2 = 15; 15 3 = (mod 26). This is t. N matches 13; 13 2 = 11; 11 3 = 33 7 (mod 26). This is h. Chapter 15: Affine Ciphers W109

118 (Text pages ) 9. b. Abraham Lincoln The clues tell us that i (8) is encrypted as S (18), so 18 m 8 + b (mod 26). a (0) is encrypted as G (6), so 6 m 0 + b (mod 26). t (19) is encrypted as P (15), so 15 m 19 + b (mod 26). The second equivalence tells us that 6 = b we didn t even have to subtract equivalences. We can substitute that into the first or third equivalence. The first is a bad choice because the 8 doesn t have an inverse. We ll use the third equivalence: 15 m (mod 26) 9 m 19 (mod 26) To solve, multiply both sides by 11 (the inverse of 19) 9 11 m (mod 26) 1 99 m (mod 26) 21 = m The encryption key was (21, 6). To decrypt, subtract 6, then multiply by 5 (the inverse of 21). Samples: Y matches 24; 24 6 = 18; 18 5 = (mod 26). This is m. U matches 20; 20 6 = 14; 14 5 = (mod 26). This is s. K matches 10; 10 6 = 4; 4 5 = (mod 26). This is u. W110 Chapter 15: Affine Ciphers

119 (Text page 151) 10. a. Guess a few letters of Peter and Tim s note. Then solve two equivalences to find m and b. Show your work. b. Decrypt Peter and Tim s note. You might suspect that the note was signed by Peter and Tim. This gives several letters: 24 8m + b (mod 26) (guessing i is encrypted as Y) (17 13m + b (mod 26)) (guessing n is encrypted as R) 7 5m (mod 26) 7 21m (mod 26) 7 5 = 21 5 mod 26 (Multiply both sides by 5, the inverse of 21.) 35 1m (mod 26) 9 1m (mod 26) so m = (9) + b (mod 26) Substitute m = 9 into first equivalence b (mod 26) 48 b (mod 26) 4 b (mod 26) so b = 4 encryption key = (9, 4) To decrypt, subtract 4, then multiply by 3. Chapter 15: Affine Ciphers W111

120 (Text page 151) 11. Each of the following was encrypted with an affine cipher. Use letter frequencies or any other information to figure out a few of the letters. Write equivalences using the letter substitutions. Solve the equivalences to find the key (m, b). Then decrypt. a. Benjamin Franklin Most common letters in the ciphertext are T, O, Y, R, P. A good first guess is that T is e. But then TA in the third line would be a 2-letter word that begins with e. That is unlikely. Another good guess is that O is e. Then TPO might be the. (Students might also suggest TPO is are, but that doesn t work.) If T is t, then 19 19m + b (mod 26) If O is e, then 14 4m + b (mod 26) 5 15m (mod 26) m (mod 26) (7 is the inverse of 15) 1 35 m (mod 26) 9 m (mod 26) Substitute m = 9 into either one of the equivalences: b (mod 26) b (mod 26) 152 b (mod 26) 4 b (mod 26) The key is (m, b) = (9, 4). To decrypt, subtract 4 and multiply by 3. Samples: B is 1; 1 4 = 3; 3 3 = 9 17 (mod 26). This is r. I is 8; 8 4 = 4; 4 3 = 12. This is m. N is 13; 13 4 = 9; 9 3 = 27 1 (mod 26). This is b. W112 Chapter 15: Affine Ciphers

121 (Text page 151) 11. b. George Bernard Shaw Most common letters in ciphertext: J, B, C, O. A good guess is that J is e. The one letter word R could be i. This gives two equivalences: 9 m 4 + b (mod 26) [1] 17 m 8 + b (mod 26) [2] Both these equivalences have even coefficients so we can t solve with inverses. We ll try another letter substitution. The KK in line 3 could be ll. Then K is l. The equivalence for this is 10 m 11 + b [3] Using [3] and [1] we have 10 m 11 + b (mod 26) ( 9 m 4 + b (mod 26)) 1 7 m (mod 26) m (mod 26) (The inverse of 7 is 15) 1 15 m (mod 26). [Solution for problem 11b continued on page W114.] Chapter 15: Affine Ciphers W113

122 Substitute m = 15 in [3] to solve for b: b (mod 26) b (mod 26) 155 b (mod 26) 1 b (mod 26). The encryption key is (15, 1). To decrypt, subtract 1 and multiply by 7 (the inverse of 15). Sample: Y is 24; 24 1 = 23; 23 7 = (mod 26). This is f. X is 23; 23 1 = 22; 22 7 = (mod 26). This is y. W114 Chapter 15: Affine Ciphers

123 Chapter 16: Finding Prime Numbers (Text page 158) 1. Find whether the following are prime numbers. Explain how you know. a. 343 divisible by 7; composite b not divisible by any prime (only have to check primes up to 32); prime c = 37 2 ; composite d not divisible by any prime (only have to check primes up to 49); prime e = 31 83; composite f = 19 53; composite Chapter 16: Finding Prime Numbers W115

124 (Text page 160) 2. Follow the steps for the Sieve of Eratosthenes to find all prime numbers from 1 to 50. The Sieve of Eratosthenes a. Cross out 1 since it is not prime. b. Circle 2 since it is prime. Then cross out all remaining multiples of 2, since they can t be prime. (Why not?) c. Circle 3, the next prime. Cross out all remaining multiples of 3, since they can t be prime. d. Circle the next number that hasn t been crossed out. It is prime. (Why?) Cross out all remaining multiples of that number. e. Repeat Step D until all numbers are either circled or crossed out. Prime Numbers from 1 to 50 Primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, As you followed the steps in Problem 2, you probably found that the multiples of the bigger prime numbers had already been crossed out. What was the largest prime whose multiples were not already crossed out by smaller numbers? 7 W116 Chapter 16: Finding Prime Numbers

125 (Text page 160) 4. a. Use the Sieve of Eratosthenes to find all primes between 1 and 130. Each time you work with a new prime, write in Table 2 the first of its multiples not already crossed out by a smaller prime. Finding a Pattern Example: When the prime is 3, the first multiple to consider is 6, but 6 has already been crossed out. Therefore, 9 is the first multiple of 3 not already crossed out by a smaller prime. Finding Primes 1 to 130 Primes less than 130: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 117, 127 Chapter 16: Finding Prime Numbers W117

126 (Text page 160) 4. b. Look at your table from 4a. Describe a pattern that tells, for any prime number, its first multiple not already crossed out by smaller prime numbers. The square of the prime number. c. When sieving for primes between 1 and 130, what was the largest prime whose multiples were not already crossed out by smaller numbers? 11 d. After you had crossed out the multiples of enough primes, you could stop because only prime numbers were left. When did this happen? After crossing out multiples of 11, only prime numbers were left. (The first multiple of 13 not crossed out would be 13 2 = 169, but that is greater than 130.) 5. a. Suppose that you used the sieve method to find the primes between 1 and 200. List the primes whose multiples you would have to cross out before only primes were left. Explain why. 15, So we only have to cross out multiples of primes less than 15. These are 2, 3, 5, 7, 11, and 13. b. Suppose that you used the sieve method to find the primes between 1 and List the primes whose multiples you would have to cross out before only primes were left. = 31.62, so we only have to cross out multiples of primes up to 31. These are 2, 3, 5, 7, 11, 13, 17, 23, 29, and 31. W118 Chapter 16: Finding Prime Numbers

127 (Text page 163) 6. a. One attempt at a formula to generate prime numbers is n 2 n Evaluate the formula for n = 0, 1, 2, 3, 4, 5. Do you always get a prime? n = 0 ; n 2 n + 41 = = 41 prime n = 1; = 41 prime n = 2; = 43 prime n = 3; = 47 prime n = 4; = 53 prime n = 5; = 61 prime b. Challenge. Find an n less than 50 for which the formula in 6a does not generate a prime. n = 41; = 41 2 = 1681 not prime 7. Look back at your list of primes. Find all pairs of twin primes between 1 and , 5 5, 7 11, 13 17, 19 29, 31 41, 43 59, 61 71, Find the Mersenne numbers for n = 5, 6, 7, and 11. Which of these are prime? = 31; prime = 63; composite (63 = 7 9) = 127; prime (not divisible by any primes only have to check primes up to 13) = 2047; composite (2047 = 23 89) Chapter 16: Finding Prime Numbers W119

128 (Text page 163) 9. Find at least three Sophie Germaine primes other than 2, 3, and 5. Answers may vary. Samples are given. 11, since = 23 23, since = 47 29, since = Challenge. Find a large prime number. (You decide whether it is large enough to please you.) Explain how you chose the number and how you know it is prime. Check students work. Some students may start with large numbers that look prime or they might limit their search to numbers that could be Mersenne primes or Sophie Germaine primes by looking at numbers of the form 2 p 1 or 2p + 1, where p is prime. Then they should test whether the number is prime. After they have found a large prime, they should explain how they tested the number. The first few Mersenne primes have exponents p = 2, 3, 5, 7, 13, 17, 31. For example, when p = 13, the Mersenne number is = Determined students could check that this is prime; since = 90.5, this would require checking all primes less than 90 to see whether they are factors of It would be time consuming, but not impossible to check that p = 17 gives a Mersenne prime. However, for p = 31, although the Mersenne number = 2,147,483,647 has been show to be prime, it is too large for students to check on a calculator. W120 Chapter 16: Finding Prime Numbers

129 (Text page 164) 11. a. Test the Goldbach Conjecture: Pick several even numbers greater than 2 and write each as the sum of two primes. (Don t use 1 in your sums, since 1 is not prime.) 10 = = = = b. Find a number that can be written as the sum of two primes in more than one way. Answers may vary. Sample: 36 = = Chapter 16: Finding Prime Numbers W121

130 Continue to the next chapter. W122 Chapter 16: Finding Prime Numbers

131 Chapter 17: Raising to Powers (Text page 169) 1. Compute the following. Reduce before your numbers get too large. a mod = (mod 1000) = (mod 1000) (mod 1000) 976 (mod 1000) b mod = (mod 1000) = (mod 1000) (mod 1000) 557 (mod 1000) c mod = (mod 1000) = (mod 1000) (mod 1000) 193 (mod 1000) d mod = (mod 1000) = (mod 1000) (mod 1000) 184 (mod 1000) Chapter 17: Raising to Powers W123

132 (Text page 170) 2. a. How many multiplications would it take to compute mod 55 using the method of repeated squaring? 18 2 = = = = = It would take 5 multiplications. b. How many multiplications would it take to compute mod 55 by multiplying 18 by itself over and over? 31 c. Compute mod 55 using the method from 2a or 2b that uses the fewest multiplications. (You can reuse calculations from this chapter.) In the book, we computed (mod 55) = (mod 55) 676 (mod 55) 16 (mod 55) W124 Chapter 17: Raising to Powers

133 (Text page 170) 3. Use the method of repeated squaring to compute each number. a. 6 8 mod 26 b. 3 8 mod = 6 6 = (mod 26) 6 4 = (mod 26) 100 (mod 26) 22 (mod 26) 6 8 = (mod 26) 484 (mod 26) 16 (mod 26) 3 2 = 3 3 = 9 4 (mod 5) 3 4 = (mod 5) 16 (mod 5) 1 (mod 5) 3 8 = (mod 5) 1 (mod 5) Chapter 17: Raising to Powers W125

134 (Text page 170) 3. c mod = 9 9 = 81 4 (mod 11) d mod = (mod 11) 16 (mod 11) 9 8 = (mod 11) 256 (mod 11) 3 (mod 11) 9 16 = (mod 11) 3 3 (mod 11) 9 (mod 11) 4 2 = 16 7 (mod 9) 4 4 = (mod 9) 49 (mod 9) 4 (mod 9) 4 8 = (mod 9) 16 (mod 9) 7 (mod 9) 4 16 = (mod 9) 49 (mod 9) 4 (mod 9) W126 Chapter 17: Raising to Powers

135 (Text page 171) 4. Use some of the powers already computed in the text to find each value. a mod = (mod 55) 1764 (mod 55) 4 (mod 55) b mod = (mod 55) 1116 (mod 55) 16 (mod 55) c mod = (mod 55) 936 (mod 55) 1 (mod 55) Chapter 17: Raising to Powers W127

136 (Text page 171) 5. a. Make a list of the values 9 n mod 55 for n = 1, 2, 4, 8, and 16. Reduce each expression (mod 55) 9 2 = (mod 55) 9 4 = (mod 55) 676 (mod 55) 16 (mod 55) 9 8 = (mod 55) 256 (mod 55) 36 (mod 55) 9 16 = (mod 55) 1296 (mod 55) 31 (mod 55) b. Combine your answers from 5a to compute 9 11 mod = (mod 55) = (mod 55) c. Combine your answers from 5a to compute 9 24 mod = (mod 55) 1116 (mod 55) 16 (mod 55) W128 Chapter 17: Raising to Powers

137 (Text page 171) 6. a. Make a list of the values 7 n mod 31, for n = 1, 2, 4, 8, and 16. Reduce each expression (mod 31) 7 2 = (mod 31) 7 4 = (mod 31) 324 (mod 31) 14 (mod 31) 7 8 = (mod 31) 196 (mod 31) 10 (mod 31) 7 16 = (mod 31) 100 (mod 31) 7 (mod 31) b. Combine your answers from 6a to compute 7 18 mod = (mod 31) 126 (mod 31) 2 (mod 31) c. Combine your answers from 6a to compute 7 28 mod = (mod 31) 980 (mod 31) 19 (mod 31) Chapter 17: Raising to Powers W129

138 Continue to the next chapter. W130 Chapter 17: Raising to Powers

139 Chapter 18: The RSA Cryptosystem (Text pages 178 and 180) 1. Use Tim s RSA public encryption key (55, 7) to encrypt the word fig. (First change the letters to numbers using a = 0, b = 1, c = 2, etc.) Encrypt using C = m 7 mod 55 f is 5; = 5 7 mod (mod 55) 25 (mod 55) i is 8; 8 7 mod (mod 55) 2 (mod 55) g is 6; 6 7 mod (mod 55) 41 (mod 55) ***Return to Text*** 2. Review: Show that 4 23 mod 55 = is too big for the calculator so use repeated squaring. 4 2 = = = = (mod 55) 4 8 = (mod 55) 1296 (mod 55) 31 (mod 55) Answer: 25, 2, = (mod 55) 961 (mod 55) = 26 (mod 55) 4 23 = (mod 55) (mod 55) = 9 (mod 55) Chapter 18: The RSA Cryptosystem W131

140 (Text page 180) 3. Dan encrypted a word with Tim s encryption key (n, e) = (55, 7). He got the numbers 4, 0, 8. Use Tim s decryption key d = 23 to decrypt these numbers and get back Dan s word. (Hint: You can use your result from Problem 2.) To decrypt, compute C d mod n = C 23 mod 55 for C = 4, C = 0, and C = 8. C = 4; 4 23 mod 55 = 9 (from Problem 2) C = 0; 0 23 mod 55 = 0 This is j. This is a. C = 8; 8 23 mod is too big for the calculator, so use repeated squaring: 8 2 = 64 9 (mod 26) 8 4 = = 9 9 = (mod 55) 8 8 = = = (mod 55) 8 16 = = = (mod 55) 8 23 = = = (mod 55) This is r. Answer: j a r W132 Chapter 18: The RSA Cryptosystem

141 Chapter 19: Revisiting Inverses in Modular Arithmetic (Text page 186) 1. For each of the following, determine whether the inverse exists in the given modulus. If it exists, use either Jenny s method or Evie s method to find it. a. 10 (mod 13) 10 is relatively prime to 13, so the inverse exists = = = 40 1 (mod 13) b. 10 (mod 15) 10 is not relatively prime to 15. c. 7 (mod 21) Answer: 4 Answer: Answer: no inverse no inverse Chapter 19: Revisiting Inverses in Modular Arithmetic W133

142 (Text page 186) 1. d. 7 (mod 18) Check numbers relatively prime to 18: 18 = so inverses can not have 2 or 3 as factors. 7 5 = = = = 91 1 (mod 18) e. 11 (mod 24) Check numbers relatively prime to 24: 11 5 = = = (mod 24) f. 11 (mod 22) Answer: 13 Answer: 11 Answer: no inverse W134 Chapter 19: Revisiting Inverses in Modular Arithmetic

143 (Text page 186) 2. Find the inverse of each of the following numbers in the given modulus. a. 11 (mod 180) Checking all numbers relatively prime to 180 would take a long time. Instead, we ll list numbers congruent to 1 (mod 180) and check whether they are 11 d for some d. (This is Evie s method from the text.) = 181 not divisible by = 361 not divisible by = 541 not divisible by = 721 not divisible by = 901 not divisible by = 1081 not divisible by = 1261 not divisible by = 1441 = Since = (mod 180), 11 and 131 are inverses mod 180. Answer: 131 b. 9 (mod 100) Numbers congruent to 1 mod 100 are 101, 201, 301, 401,. Check these to find whether 9 divides them. We find that 9 89 = (mod 100). Answer: 89 c. 7 (mod 150) Numbers congruent to 1 (mod 150) are 151, 301, 451, 601,. Check to find which of these is divisible by 7. We find that 7 43 = (mod 150). Answer: 43 Chapter 19: Revisiting Inverses in Modular Arithmetic W135

144 Continue to the next chapter. W136 Chapter 19: Revisiting Inverses in Modular Arithmetic

145 Chapter 20: Sending RSA Messages (Text pages 189 and 192) Class Activity Follow the instructions in the text to choose your own RSA key. Use your own paper to record your calculations. Then record your encryption key here. RSA encryption: n = e =. Answers will vary. Record your decryption key d and your primes p and q in a secret place so you won t forget it. (If you put it here, other people can decrypt messages sent to you. But if you lose it even you won t be able to decrypt.) ***Return to Text*** 1. Use Dan s keyword CRYPTO to decrypt his Vigenère message to Tim. Dan Chapter 20: Sending RSA Messages W137

146 (Text page 192) 2. Here is the reply Tim sent to Dan: a. Dan s RSA decryption key is d = 5. Use it to find the keyword that Tim encrypted. (Tim used Dan s encryption key (n,e) = (221, 77).) Show your work. Use the next page if you need more space. Compute C 5 mod 221 for C = 32, 209, 165, and 140: C = 32; 32 5 mod 221 = mod 221 C = 209; 2 This is C mod 221 (Too big for calculator; use repeated squaring.) = (mod 221) = (mod 221) (mod 221) = 14 mod 221 This is O. W138 Chapter 20: Sending RSA Messages

147 Use this space to continue your work from 2a. C = 165; mod 221 (Too big for calculator.) = (mod 221) C = 140; = (mod 221) (mod 221) 3 (mod 221) This is D mod 221 (Too big for calculator.) = (mod 221) = (mod 221) (mod 221) 4 This is E. The keyword is C O D E. Chapter 20: Sending RSA Messages W139

148 (Text page 192) 2. b. Use the keyword you found in 2a to decrypt the Vigenère message Tim sent to Dan. 3. Follow the instructions in the text to combine RSA with the Vigenère cipher and send an RSA message to someone. Use your own paper to write your message and record your calculations. W140 Chapter 20: Sending RSA Messages